Optimal. Leaf size=108 \[ \frac {a (3 A+2 C) \sin (c+d x)}{3 d}+\frac {a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+3 C)+\frac {a C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {a C \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3034, 3023, 2734} \[ \frac {a (3 A+2 C) \sin (c+d x)}{3 d}+\frac {a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+3 C)+\frac {a C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {a C \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2734
Rule 3023
Rule 3034
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos (c+d x) \left (4 a A+a (4 A+3 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a C \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos (c+d x) (4 a (3 A+2 C)+3 a (4 A+3 C) \cos (c+d x)) \, dx\\ &=\frac {1}{8} a (4 A+3 C) x+\frac {a (3 A+2 C) \sin (c+d x)}{3 d}+\frac {a (4 A+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a C \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 77, normalized size = 0.71 \[ \frac {a (24 (4 A+3 C) \sin (c+d x)+24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+8 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 76, normalized size = 0.70 \[ \frac {3 \, {\left (4 \, A + 3 \, C\right )} a d x + {\left (6 \, C a \cos \left (d x + c\right )^{3} + 8 \, C a \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 3 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (3 \, A + 2 \, C\right )} a\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.45, size = 86, normalized size = 0.80 \[ \frac {1}{8} \, {\left (4 \, A a + 3 \, C a\right )} x + \frac {C a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {C a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (A a + C a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a + 3 \, C a\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.20, size = 96, normalized size = 0.89 \[ \frac {a C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 90, normalized size = 0.83 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 96 \, A a \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.71, size = 212, normalized size = 1.96 \[ \frac {\left (A\,a+\frac {3\,C\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (5\,A\,a+\frac {49\,C\,a}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (7\,A\,a+\frac {31\,C\,a}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (3\,A\,a+\frac {13\,C\,a}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+3\,C\right )}{4\,\left (A\,a+\frac {3\,C\,a}{4}\right )}\right )\,\left (4\,A+3\,C\right )}{4\,d}-\frac {a\,\left (4\,A+3\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.10, size = 226, normalized size = 2.09 \[ \begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {3 C a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 C a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________